Problem 108 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{8}{100}$$

Step-by-Step Solution

Answer

The fraction \( \frac{8}{100} \) reduces to \( \frac{2}{25} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{8}{100} \), we need to find the greatest common divisor (GCD) of the numerator (8) and the denominator (100). The GCD is the largest number that divides both without leaving a remainder.

Step 2: Calculate the GCD

Check the divisibility of both numbers starting from the largest factors. We find that both 8 and 100 are divisible by 4, as their remainders are zero. In fact, 4 is the largest number that can divide both, hence the GCD is 4.

Step 3: Divide Both Numerator and Denominator by the GCD

Take the fraction \( \frac{8}{100} \) and divide both the numerator and the denominator by the GCD, which is 4: \[ \frac{8 \div 4}{100 \div 4} = \frac{2}{25} \]

Step 4: Verify the Simplified Fraction

Ensure that the new fraction, \( \frac{2}{25} \), is in its simplest form. Check if there is any common factor between 2 and 25 besides 1. Since there are none, \( \frac{2}{25} \) is indeed the simplest form.

Key Concepts

Greatest Common Divisor (GCD)

Before reducing a fraction, it's crucial to determine the greatest common divisor (GCD) of its numerator and denominator. The GCD is the largest number that divides both numbers neatly, i.e., it leaves no remainder.
Finding the GCD involves:

• Identifying factors: Listing all factors, or numbers that divide evenly into both the numerator and denominator.
• Comparing factors: Ensuring these numbers are the largest possible common ones.

Using GCD efficiently simplifies fractions without unnecessary trial and error. In our example, for the fraction \( \frac{8}{100} \), the numbers 2 and 4 can divide both 8 and 100, but 4 is the greatest. Thus, the GCD is 4.

Lowest Terms

Expressing a fraction in its lowest terms means reducing it as much as possible until no further division is possible by numbers other than 1.
This is achieved by dividing both the numerator and the denominator by their greatest common divisor.
For instance, with \( \frac{8}{100} \):

• Divide the numerator (8) by the GCD (4) to get 2.
• Divide the denominator (100) by the GCD (4) to get 25.
• The fraction becomes \( \frac{2}{25} \).

The fraction \( \frac{2}{25} \) is now in its lowest terms, ensuring it's as simple as possible, making it easier to understand and work with.

Simplified Fractions

When we talk about simplified fractions, we're discussing fractions that are in their simplest form. This means:

• No number other than 1 can evenly divide both the numerator and the denominator.
• The process involves reducing the fraction using the greatest common divisor.
• Verification to ensure that the simplified fraction cannot be reduced further.

Using our exercise example, once we've divided \( \frac{8}{100} \) by 4, we obtain \( \frac{2}{25} \). Testing for further reduction confirms 2 and 25 share no common factors aside from 1.
Simplified fractions are beneficial for clearer representation of ratios, easier calculations, and precise communication of quantities.